The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background

TL;DR

This paper investigates early-time asymptotics of linear wave solutions on Schwarzschild spacetime, revealing logarithmic terms in their expansions and proposing an alternative approach to late-time behavior analysis without smooth or compact data assumptions.

Contribution

It derives new early-time asymptotics for higher angular modes of linear waves on Schwarzschild backgrounds, highlighting logarithmic terms and deviations from Price's law.

Findings

Logarithmic terms appear at specific orders in asymptotic expansions.

Different boundary conditions lead to distinct asymptotic behaviors.

Proposes an approach to study late-time asymptotics without smooth data assumptions.

Abstract

In this paper, we derive the early-time asymptotics for fixed-frequency solutions $\phi_\ell$ to the wave equation $\Box_g \phi_\ell=0$ on a fixed Schwarzschild background ($M>0$) arising from the no incoming radiation condition on $\mathcal I^-$ and polynomially decaying data, $r\phi_\ell\sim t^{-1}$ as $t\to-\infty$, on either a timelike boundary of constant area radius (I) or an ingoing null hypersurface (II). In case (I), we show that the asymptotic expansion of $\partial_v(r\phi_\ell)$ along outgoing null hypersurfaces near spacelike infinity $i^0$ contains logarithmic terms at order $r^{-3-\ell}\log r$. In contrast, in case (II), we obtain that the asymptotic expansion of $\partial_v(r\phi_\ell)$ near spacelike infinity $i^0$ contains logarithmic terms already at order $r^{-3}\log r$ (unless $\ell=1$). These results suggest an alternative approach to the study of late-time asymptotics near future timelike infinity $i^+$ that does not assume conformally smooth or compactly supported Cauchy data: In case (I), our results indicate logarithmic modifications to Price's law for each $\ell$-mode. On the other hand, the data of case (II) lead to much stronger deviations from Price's law. In particular, we conjecture that compactly supported scattering data on $\mathcal H^-$ and $\mathcal I^-$ lead to solutions that exhibit the same late-time asymptotics on $\mathcal I^+$ for each $\ell$: $r\phi_\ell|_{\mathcal I^+}\sim u^{-2}$ as $u\to\infty$.